We present an interface between the multipurpose Monte Carlo tool MadGraph5_aMC@NLO and the automated amplitude generator GoSam. As a first application of this novel framework, we compute the NLO corrections to \(pp \rightarrow \) \(t\bar{t}H\) and \(pp \rightarrow \) \(t\bar{t}\gamma \gamma \) matched to a parton shower. In the phenomenological analyses of these processes, we focus our attention on observables which are sensitive to the polarisation of the top quarks.

Spin polarisation of \(t\bar{t}\gamma \gamma \) production at NLO+PS with GoSam interfaced to MadGraph5_aMC@NLO

Eur. Phys. J. C
Spin polarisation of t t¯γ γ production at NLO+PS with GOSAM interfaced to MADGRAPH5_AMC@NLO
Hans van Deurzen 2
Rikkert Frederix 1
Valentin Hirschi 0
Gionata Luisoni 2
Pierpaolo Mastrolia 2 5
Giovanni Ossola 3 4
0 SLAC, National Accelerator Laboratory , 2575 Sand Hill Road, Menlo Park, CA 94025-7090 , USA
1 Physik Department T31, Technische Universität München , James-Franck-Str. 1, 85748 Garching , Germany
2 Max-Planck-Institut für Physik , Föhringer Ring 6, 80805 Munich , Germany
3 The Graduate School and University Center, City University of New York , 365 Fifth Avenue, New York, NY 10016 , USA
4 New York City College of Technology, City University of New York , 300 Jay Street, Brooklyn, NY 11201 , USA
5 Dipartimento di Fisica e Astronomia, Università di Padova and INFN Sezione di Padova , via Marzolo 8, 35131 Padua , Italy
We present an interface between the multipurpose Monte Carlo tool MadGraph5_aMC@NLO and the automated amplitude generator GoSam. As a first application of this novel framework, we compute the NLO corrections to pp → t t¯H and pp → t t¯γ γ matched to a parton shower. In the phenomenological analyses of these processes, we focus our attention on observables which are sensitive to the polarisation of the top quarks.
1 Introduction
The development of automated tools for precise
calculations of total cross sections and differential distributions in
high-energy collisions has undergone a dramatic
acceleration in the last decade. While Leading-Order (LO) tools,
based on automated tree-level calculations, have been
available for a long time [
1–11
], the needs of the experimental
analyses at the Large Hadron Collider (LHC) and a deeper
understanding of the structure of scattering amplitudes [
12–
14
] at one loop led to the development of several
computer frameworks for the automated computation of loop
matrix elements [
15–20
] and physical observables at
Nextto-leading-Order (NLO) accuracy [
21–27
]. Moreover,
techniques to properly deal with the merging of different
multiplicities in the final states and the matching [
28,29
] to parton
shower were successfully developed [
30–34
] and are
nowadays available.
Advanced automated calculations have been performed
by embedding the codes for generating virtual corrections at
NLO precision, the so called One Loop Providers (OLPs),
within a Monte Carlo program (MC). The interplay between
MCs programs and OLPs is controlled by means of
interfaces, which allow the user to get direct access to the main
features of the MC code, bypassing the need of knowing the
technical details of the OLP, which is ideally an inner engine
within the MC generator. Many of such interfaces are based
on the standards settled by the Binoth Les Houches Accord
(BLHA) [
35,36
], which defines specifications of the
communication between MCs and OLPs.
The LHC has recently started Run II, collecting data at
an energy scale never explored before. Within this activities
automated multipurpose tools for particle collisions
simulation are of fundamental importance for comparing theoretical
predictions with experimental data, thereby extracting
important information as regards the Standard Model (SM) and
exploring all traces of deviations from it. The need for
flexible tools which at the same time can provide accurate
predictions, both in the SM and Beyond (BSM), may become of
primary relevance in the near future. It is therefore important
to be able to connect different tools to increase the reliability
of the results.
With these goals in mind, we present the interface
between the multipurpose NLO Monte Carlo tool
MadGraph5_aMC@NLO(MG5_aMC) [
27
] and the automated
one-loop amplitude generator GoSam [
20
]. The advantage
of this combination is twofold. On the one hand, this tandem
allows the user of MG5_aMC to switch between two options
of OLPs, namely between the inhouse code MadLoop [
23
]
fully integrated directly in the MC distribution package, and
GoSam. Thus, the user can experience the evaluation of NLO
virtual corrections by means of two alternative solutions
corresponding to different algorithms and methods of generation
and evaluation of Feynman amplitudes. On the other hand,
GoSam is interfaced to several MCs codes, like Sherpa [
21
],
Herwig++ [
26
], Powheg [
25
], beside MG5_aMC,
therefore the user of the MCs can explore and compare the
different features of the event generators, without being biased
by the performances of the OLPs, since they all can be run
using GoSam.
As an illustration of the novel framework MG5_aMC +
GoSam, we present its application to the NLO corrections
to pp → t t¯H , H → γ γ and the continuum pp → t t¯γ γ
matched to a parton shower. The production of a Higgs boson
in association with a pair of top anti-top quarks is an
important process to directly study the Yukawa coupling of the
Higgs boson with massive fermions. Such a channel, and
its corresponding backgrounds, were recently the subject of
detailed studies, both at LO [
37–39
] and NLO [
40
]
precision. Very recently, new analyses have appeared which
further extend these studies including the decay of the top and
anti-top quark into bottom quarks and leptons [
41
],
considering the production of a top-quark pair in conjunction with up
to two vector bosons [
42
], and exploring the CP-structure
of the top-Higgs coupling [
43
]. In the phenomenological
analysis contained in this paper, we focus our attention on
observables sensible to the polarisation of the top quarks,
such as angular variables which involve the decay of the top
quark.
The paper is organised as follows: in Sect. 2, after a
general introduction to the GoSam and MG5_aMC codes, we
will discuss the interface between the two frameworks and its
validation. In Sect. 3, we will present an application of the
GoSam+ MG5_aMC interface, namely the study of NLO
corrections to pp → t t¯H , H → γ γ and pp → t t¯γ γ
matched to a parton shower. Finally in Sect. 4, we will draw
our conclusions.
2 Computational setup
For the computations contained in this paper, the automated
one-loop amplitude generator GoSam has been fully
interfaced to the MG5_aMC Monte Carlo framework.
In this section, we briefly review the main characteristics
of each of these tools and describe the details of the interface
between them, which allows one to use one-loop amplitudes
generated by GoSam within MG5_aMC. Finally we discuss
the validation of the interface by means of a comparison with
an independent framework.
2.1 GoSam
The main idea that distinguishes the GoSam framework [
20
]
from other codes for the automated generation of one-loop
amplitudes is the combined use of automated diagrammatic
generation and algebraic manipulation in d = 4 − 2
dimensions, thus providing analytic expressions for the integrands,
with d-dimensional integrand-level reduction techniques, or
tensorial reduction. Amplitudes are automatically generated
via Feynman diagrams and, according to the reduction
algorithm selected by the user, are algebraically manipulated and
cast in the most appropriate output [
44–49
]. The individual
program tasks are controlled by means of a python code,
while the user only needs to prepare an input card to specify
the details of the process to be calculated without worrying
about internal details of the code generation.
After the generation of all contributing diagrams, the
virtual corrections are evaluated using the integrand reduction
via Laurent expansion [
50
], provided by Ninja [
51,52
], or the
d-dimensional integrand-level reduction method [
53–55
], as
implemented in Samurai [
56,57
]. Alternatively, the
tensorial decomposition provided by Golem95C [
58–61
] is also
available. The scalar loop integrals can be evaluated using
Golem95C, OneLOop [62], or QCDLoop [63,64].
The GoSam framework can be used to generate and
evaluate one-loop corrections in both QCD and electroweak
theory [
65
]. Model files for Beyond Standard Model (BSM)
applications generated from a Universal FeynRules Output
(UFO) [
66,67
] or with LanHEP [68] are also supported. A
model file which contains the effective Higgs–gluon
couplings that arise in the infinite top-mass limit is also
available in the current distribution and it was successfully used to
compute the virtual corrections for the production of a Higgs
boson in association with 2 and 3 jets [
69,70
].
The computation of physical observables at NLO
accuracy, such as cross sections and differential distributions,
requires one to combine the one-loop results for the virtual
amplitudes obtained with GoSam, with other parts of the
calculations, namely the computation of the real emission
contributions and of the subtraction terms, needed to control
the cancellation of IR singularities. In some of the earlier
calculations performed with GoSam [
71–75
], the problem was
solved by means of an ad hoc adaptation of the
MadDipoleMadgraph4- MadEvent framework [
1,9,76,77
].
Far more efficiently, this task can be performed by
embedding the calculation of virtual corrections within a
multipurpose Monte Carlo program (MC), that can also provide
the phase-space integration, which is what is pursued in this
work. In this case, the MC takes control over the different
stages of the calculation, in particular the phase-space
integration and the event generation, and calls GoSam at runtime
to obtain the corresponding value of the one-loop amplitude
at the given phase-space points. This approach has the great
advantage of making available to the user all the advanced
features that the MC generator provides, for example to allow
for parton showering, further decays of the final-state hard
particles retaining spin correlation and merging of different
final state multiplicities.
While in the present paper we will focus on the interface
between GoSam and MG5_aMC, it is worth mentioning that
a number of phenomenological results can be found in the
literature which were obtained by combining GoSam with
other MC programs, in particular with Sherpa [
21,78–81
],
Herwig++ [
26,82
], and Powheg [
25,83,84
].
2.2 MadGraph5_aMC@NLO
The MG5_aMC framework [
27
] has been developed to be
able to generate events and compute differential cross
sections with a high-level of automation. The central idea behind
the code is that the structure of cross sections is essentially
independent of the process under consideration. Therefore,
once this structure has been set up, any cross section can be
computed within the framework. For example, even though
the matrix elements are process and theory dependent, they
can be computed from a very limited set of instructions based
on the Feynman rules.
The core of the MG5_aMC framework is based on
treelevel amplitude generation, since in this code the matrix
elements used in both LO and NLO computations are
constructed from tree-level Feynman diagrams. The generation
of these amplitudes is based on three elements which are key
to taming the complexity of the computation as the number of
external particles increases: colour decomposition, helicity
amplitudes and recycling of identical substructures between
diagrams. The internal algorithms used have been described
extensively in Ref. [
85
] and Ref. [
18,23,27,53,86,87
] for
the generation of tree-level and one-loop matrix elements,
respectively.
Beyond the lowest order in perturbation theory,
intermediate contributions to the computation of (differential)
cross sections are plagued by divergences. In particular the
soft/collinear divergences in the numerical phase-space
integration over the real-emission (Bremsstrahlung) corrections
are non-trivial to deal with. In MG5_aMC, the FKS
subtraction method [
88,89
] is used to factor out the
singularities in order to cancel them analytically with singularities
present in the virtual corrections. The FKS subtraction is
based on partitioning the phase space, in which each
phasespace region has at most one collinear and one soft
singularity. These singularities are subtracted before performing the
numerical phase-space integration by using a straightforward
plus-description. The subtraction terms have been integrated
analytically, using dimensional regularisation, once and for
all, resulting in explicit poles in 1/ and 1/ 2, which cancel
similar poles in the virtual corrections and the PDFs.
Independently of which code one uses for the
computation of the virtual corrections, for example MadLoop [
23
] or
GoSam [
20
], optimisations in the phase-space integration of
these contributions are used, as described in Ref. [
27
]. That
is, during the phase-space integration an approximation of
the virtual corrections based on the born matrix elements is
created dynamically. These approximate virtual matrix
elements are very fast to evaluate and can therefore be
efficiently integrated numerically with high statistics. The small
difference between the approximate and the exact virtual
corrections is relatively slow to evaluate for a given
phasespace point, but given that this is a very small contribution
to the final result, the requirement on the relative
precision with which it needs to be computed can be relaxed.
Therefore, using low statistics for this complicated
contribution suffices, greatly reducing the overall computational
time.
To match the short-distance matrix elements to a
parton shower, the framework of MG5_aMC employs the
MC@NLO technique [
28
] available for Pythia 6 [
90
],
Herwig 6 [
91
], Pythia 8 [
92
] and Herwig++ [
93
]. In this method,
the possible double counting between the NLO corrections
and the parton shower is accounted for by explicitly
subtracting the parton shower approximation for the emission
of a hard parton from the real-emission contributions, and
the parton shower approximation of a non-emitting parton
from the virtual corrections. To consistently merge various
multiplicities at NLO accuracy and match them to the parton
shower, the FxFx merging method [
33
] is available.
2.3 Interface
The interface between GoSam and MG5_aMC is based on
the standards of the first BLHA defined in [
35
]. When running
the MG5_aMC interactive session, the command
$ set OLP GoSam
changes the employed OLP from its default MadLoop to
GoSam. Alternatively, the file
input/mg5_configuration.txt
can be directly edited to include the line OLP = GoSam.
The BLHA order and contract file system allows for a
basic communication between the two codes to exchange
the most fundamental information about the number and
type of subprocesses, the powers of the couplings involved
in the specific process, the schemes in which the
computation should be performed and also the value of parameters
like masses and widths. For static parameters, which do not
change at each phase-space point, but stay constant during
the MC integration and event generation, the information is
passed via a SUSY Les Houches Accord (SLHA) parameter
file. This is created by MG5_aMC and read by GoSam. The
#OLE_order written by MadGraph5_aMC@NLO
path to this SLHA file is specified in the order file with the
key word ModelFile. An example generated for the
computation of t t¯γ γ is shown in Fig. 1. For parameters which
instead may change at each phase-space point, the BLHA
1 interface defines an array to pass the numerical value of
dynamical variables. The definition and order of the
parameters passed through this array is set in the order file using the
keyword Parameters. Although in principle extensible to
up to ten parameters, at present only the first entry is used,
to communicate the value of αS.
Further customisation of the one-loop amplitudes which
need to be generated can be achieved by editing a separate
input file for GoSam. In the file gosam.rc, additional
information can be specified, i.e. the model, the particle content
of the loop diagrams, and the number of active flavours.
Here it is also possible to define ad hoc filters to remove
unwanted diagrams or loop contributions which are known
to be negligible or vanishing, which may, however, if kept
in the calculation, introduce numerical instabilities or slow
down the evaluation. Some settings are present by default,
but many more can be introduced by the user. We refer to
the appendix for a more extensive list of possible options.
The GoSam input file needs to be edited by hand and can be
found at
Template/loop_material/OLP_specifics/GoSam/
gosam.rc,
in the MG5_aMC repository, or at
OLP_virtuals/gosam.rc,
in the folder that is generated by MG5_aMC automatically
when a new process is started. After the input file is ready,
any NLO process can be generated following the general
MG5_aMC procedure.
The interface between MG5_aMC and GoSam is
available starting from MG5_aMC version 2.3.2.2. The two codes
can be downloaded from the following URL:
MG5_aMC: https://launchpad.net/mg5amcnlo
GoSam: http://gosam.hepforge.org/
2.4 NLO predictions and validation
To validate the interface, and consequently the results we
present in Sect. 3, several cross checks were performed.
The loop amplitudes of GoSam and MadLoop were
compared for single phase-space points and also at the level
of the total cross section for a number of different
processes, as presented in a dedicated table in [
94
]. Furthermore,
for pp → t t¯γ γ , a fully independent check was also
performed by computing the same cross section using GoSam
interfaced to Sherpa. The comparison between the results
obtained with MG5_aMC+GoSam, Sherpa+GoSam, and
MG5_aMC+MadLoop is presented Table 1, where we
report the total integrated cross sections for LO and NLO
at a centre-of-mass energy √s = 8 TeV.
The results shown in this and the following sections are
computed using the following setup. The mass of the Higgs
was set to m H = 125 GeV, the mass of the top quark to
mt = 173.2 GeV. We work in the N f = 5 model. The
value of the electroweak coupling is set to its low energy
limit α−EW1 = 137.0. The mass of the Z boson was set to
m Z = 91.1876 GeV and the value of the Fermi constant
to G F = 1.16639 · 10−5 GeV−2, which fixes the
electroweak scheme. For the photons, we used the isolation
procedure introduced by Frixione [
95
] with minimal transverse
momentum pγ T min = 20 GeV, radius of isolation Rγ < 0.4
and Frixione parameters n = 1.0 and γ = 1.0.
Furthermore, we applied an isolation radius between the two
photons Rγ γ = 0.4. In leading order calculations, we used the
PDF set cteq6L1 [
96
]. At next-to-leading order, we instead
used the PDF set CT10. The renormalisation and
factorisa)V10−5
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σtt¯γ γ , √s = 8 TeV
LO (pb)
NLO (pb)
MG5_aMC + MadLoop
MG5_aMC + GoSam
Sherpa+GoSam
tion scales are set to μR = μF = μ0 with
3 Results
μ0 =
HˆT 1
2 = 2
final state i
m T,i
In Fig. 2, above, we compare LO and NLO predictions
for the transverse momentum of the top quark obtained with
GoSam+MG5_aMC, while below we compare the same
NLO predictions with results obtained using GoSam+Sherpa.
In Fig. 3 we do the same for the photon pair invariant mass.
All predictions are computed for a centre-of-mass energy of
8 TeV.
(1)
In this section we present results at NLO+PS level for the
LHC at 13 TeV and compare the background process t t¯γ γ ,
where the photons are directly radiated from the quarks, with
the signal process t t¯H in which the Higgs boson decays to
two photons. We will refer to the latter simply as “t t¯H ”; it
should be understood that we consider only the process with
the photonic Higgs decay t t¯H , H → γ γ . As a reference, we
also include results for the total cross sections and a selection
of distributions obtained at 8 TeV.
The study is performed using NLO predictions for t t¯H and
continuum t t¯γ γ production. The top and anti-top quarks are
√s = 8 TeV
LO (pb)
NLO (pb)
K -factor
√s = 13 TeV
LO (pb)
NLO (pb)
K -factor
NLO ratio 13TeV/8TeV
subsequently decayed semi-leptonically t → W +(→ l¯ νl )b,
t¯ → W −(→ l, ν¯l )b¯ with MadSpin [
97,98
], taking into
account spin correlation effects, and then showered and
hadronised by means of Pythia 8.2, using its default
parameters, but with underlying event turned off. The short-distance
events were generated and compared with two slightly
different sets of cuts in order to verify that they had no impact
on the results at the level of the analysis. Apart from the
kinematical requirements on the reconstructed objects, which
we will describe below, we use identical model parameters,
renormalisation and factorisation scales, PDFs and photon
isolation as described in Sect. 2.4.
Note that for the background process we neglect effects of
photon bremsstrahlung from the charged top decay products,
which can at least partially be reduced by applying proper
kinematical cuts. For the spin correlation observables, on
which we will focus our attention in the last part of this
section, a LO study [
37
] showed that the impact of neglecting
these contributions is present but not dramatic.
The analysis cuts are designed to increase the signal over
the background, but are by no means optimised to maximise
the enhancement. The two photons from the Higgs decay (or
the two hard photons in the t t¯γ γ process) are required to be
isolated and fulfill
pT,γ > 20 GeV, |ηγ | < 2.5,
123 GeV < mγ γ < 129 GeV,
where the invariant mass requirement selects a window
around the Higgs boson mass, which reduces the background
significantly without altering the signal strength.
Furthermore, we require the events to have two oppositely charged
leptons and two b-jets coming from the top and anti-top
decays. The leptons are selected requiring
following the cross section correspond to the scale and PDF variations,
respectively. We also report the ratio between the cross sections at the
two centre-of-mass energies
pT,l± > 10 GeV, |ηl± | < 2.7.
2.90(1) · 10−7 +30 % +14 %
3.71(1) · 10−7 +−−4821%%%−+−115615%%%
1.28(1)
pp → tt¯H , H → γ γ
The b-jets are defined to be jets containing at least one
lowest lying B meson. The jets themselves are defined by
clustering all stable hadrons and photons, but excluding the
two photons selected using Eq. (2), using the anti-kT
algorithm as implemented in the code FastJet [
99–101
], with
We use MC truth information to select the photons from
the Higgs decay (signal) or hard events (background) as well
as the leptons and b-jets coming from the top and anti-top
decays. As reconstructed (anti-)top quark, we use the
fourmomentum of the (anti-)top quark just before it decays, as
provided in the Pythia 8 event record. In the presence of these
analysis cuts, we obtain the cross sections reported in Table 2.
In the next section we focus our attention on some relevant
observables related to a single particle, whereas in Sect. 3.2
we will concentrate on observables which can directly probe
correlation effects due to the top quark polarisation.
In the following, unless specified otherwise, the plots will
always consist of four distributions. The top curves show the
differential cross sections for a given observable and for both
the signal and the background process. The two middle insets
display the relative uncertainty of the t t¯H and t t¯γ γ
predictions, respectively. The scale dependence (transparent band)
is estimated by taking the envelope of the nine predictions
obtained by the separate variation of renormalisation and
factorisation scales by factors of 0.5 and 2 around the central
scale μ0 defined in Eq. (1). For estimating the PDF
uncertainty (dotted lines), we use the Hessian method. Finally, the
bottom inset highlights the differential signal-to-background
ratio.
3.1 Single particle observables We start comparing the transverse momentum distribution of the reconstructed top and anti-top quarks, which is shown in
Top quark transverse momentum
Top quark transverse momentum
Figs. 4 and 5, for a centre-of-mass energy of 8 and 13 TeV,
respectively.
For either the signal or background process, the shapes of
the top and anti-top quark pT distributions are very similar,
as expected, and therefore the same is true for the
signalto-background ratio. In both cases, it reaches a maximum
between 50 and 100 GeV and than decreases slightly in
the high transverse momentum tail. Increasing the
centreof-mass energy from 8 to 13 TeV does not lead to significant
changes. Since this is true also for the other distributions that
we studied, in the following we will only report and comment
on the 13 TeV scenario.
It is worth noting, by looking at Fig. 5, that the uncertainty
due to the PDF variation is larger in t t¯H , due to the dominant
gluon-channel production, and is increasing for larger
transverse momenta. At pT ≈ 400 GeV the PDF uncertainty for
t t¯H is around 20 %, whereas it stays below 15 % for t t¯γ γ .
LHC 13 TeV
Reconstructed Higgs boson transverse momentum
NLO tt¯H
PDF variation
NLO tt¯γγ
PDF variation
Figure 6 shows the transverse momentum and the rapidity
of the photon pair, which for the signal process corresponds
to the one of the reconstructed Higgs boson. Since in t t¯γ γ
the photon pair does not originate from a massive particle
decay, its transverse momentum is softer and the spectrum
falls off faster for large pT . The signal-to-background
rapidity curve shows that photons coming from the decay of the
Higgs boson are generally produced more centrally. This is
not surprising given that such a decay does not feature a
collinear enhancement, contrary to the case of t t¯γ γ .
The transverse momentum distribution for the single
photons (ordered according to their pT ) is shown in Fig. 7. The
shoulder in the t t¯H signal distribution stemming from the
presence of the Higgs boson resonance is also visible in the
background shape, albeit less pronounced, given the
invariant mass cut on the photon pair. As expected, the shoulder
300 350
400
20
40 60 80 100 120 140 160 180 200
pT, γ1
Second leading photon transverse momentum
NLO tt¯H
PDF variation
NLO tt¯γγ
PDF variation
-2
-1
1
2
0
yγ2
Top quark rapidity
10−7
is shifted towards lower transverse momenta for the case of
t t¯γ γ , because of the initial state collinear enhancement.
It is particularly interesting to compare the rapidities of
the top and anti-top quarks (Fig. 8) and of their decay
products. This highlights the well known difference between the
broadness of the respective top and anti-top quark rapidity
distribution and it can be used to improve on background
discrimination. This difference is known as the charge
asymmetry and is usually quantified by the following observable:
(5)
AC σ ( |y| > 0) − σ ( |y| < 0) ,
t¯t = σ ( |y| > 0) + σ ( |y| < 0)
where |y| = |yt| − |y¯t|. This observable has been measured
at the LHC by both the ATLAS [
102,103
] and the CMS [
104–
106
] collaboration in the context of top-quark studies.
For top-pair production at the LHC AtC¯t is positive, i.e. top
quarks are produced at larger rapidities compared to anti-top
quarks [
107
]. It is, however, known that the presence of a
photon reverses the sign of AtC¯t already at tree-level [
108
].
This change in the rapidity distributions of the top and
antitop quarks, due to additional photon radiation, can clearly be
seen in the plots of Fig. 9, which compare yt and y¯t
individually for t t¯H and t t¯γ γ . In the signal process the additional
presence of a Higgs boson in the final state does not change
the qualitative result, as compared to simple t t¯-production.
This can be seen in the upper portion of the top plot in Fig. 9,
which shows that top quarks are produced at slightly higher
rapidities, as compared to anti-tops, which are more central,
leading to a positive AtC¯t. In the lower left plot, instead, the
effect is reversed. The presence of the additional photons
causes the top quarks to be more central compared to the
anti-tops.
This effect is even more visible when comparing directly
the distributions for t t¯H with the ones for t t¯γ γ . To better
appreciate the change in the shape, which is only marginally
visible in Fig. 8, we plot the same distribution normalised
to the inclusive t t¯H cross section on the right hand side of
Fig. 9. From the upper plot it becomes clear that the top quark
rapidities have very similar shape in both the signal and the
background process, although in the latter the tops are
produced at slightly higher rapidities. This means that despite
the top being produced at higher rapidities as compared to
the anti-top in t t¯H , overall, they are still slightly more
central than in t t¯γ γ . The opposite is true for the anti-top quark
rapidity, and, therefore, the difference is even more visible
in the lower right plot of Fig. 9.
Analogous conclusions can be derived by looking at the
rapidities of the top decay products. They are shown in
Figs. 10 and 11, where the rapidities of the b- and b¯-jets
and the charged leptons are shown, respectively.
In Fig. 12 we compare the rapidities of the leading and
second leading photon. Not surprisingly, the photon coming
form the Higgs boson decay are produced more centrally as
compared to the ones radiated from the partons.
3.2 Spin polarisation observables
In this section we focus on observables that allow for
investigating polarisation effects of the top and anti-top quarks as
well as in their decay products. This can be done by
studying angular variables which involve the decay products of
t and ¯t rapidity for tt¯H and tt¯γγ
b-jet rapidity
top tt¯H
anti-top tt¯H
Fig. 9 Upper plot the rapidity distribution comparison between the
top and anti-top quark for t t¯H (above) and t t¯γ γ (below). Lower plot
normalised top (above) and anti-top (below) rapidity distribution for
signal and background
the top and anti-top quarks; both top quarks are considered
to decay semi-leptonically (same as in the previous section).
We stress that a similar analysis was already performed at
LO in Ref. [
37
].
10−7
Typically, for hadronic t t¯-production, very specific
kinematic frames are defined [
109–111
]. In the following we will
consider the three-dimensional opening angle θll between the
leptonic decay products of the top (l+) and anti-top quarks
(l−), defined in three different frames. The most
straightforward possibility is to define θll in the laboratory frame
(referred to as lab frame in the following). The results for this
case are shown in Fig. 13. The definition of two other frames,
Positively charged lepton rapidity
Leading photon rapidity
Fig. 11 Rapidity distribution of the lepton (top) and anti-lepton
(bottom) produced in the decay of the top and anti-top quark, respectively
introduced for the first time in [
110
], became customary in
polarisation studies, since they capture particularly well spin
correlation effects.
For these particular frames, we define θll to be the angle
between the direction of flight of l+, measured in frame
where the top quark is at rest, and the direction of flight
of l−, measured in the frame where the anti-top quark is at
-2
-1
1
2
0
yγ2
Fig. 12 Rapidity distribution of the leading (top) and second leading
photon (bottom)
rest. Since two rest frames are involved in this definition,
a common initial frame needs to be specified, from which
the (rotation-free) Lorentz boost can be applied in order to
transform the system to the t and t¯ rest frames. We choose
two possible starting points, which we label as frame-1 and
frame-2, defined as follows:
Cosine θl+l− in the laboratory frame
Cosine θl+l− in the frame-1
Fig. 13 cos θll distribution for the signal (t t¯H ) and background (t t¯γ γ )
processes in the laboratory frame. The exact definition of the angle θ is
given in the text. The t t¯γ γ prediction is normalised to the t t¯H inclusive
cross section. In the upper plot, we compare LO with NLO predictions
and show their K -factor separately for t t¯H and t t¯γ γ in bottom insets.
In the lower plot, we show NLO relative uncertainties with the
signalto-background ratio as the last bottom inset
– frame-1 the Lorentz boosts to bring t and t¯ separately at
rest are defined with respect to the t t¯-pair centre-of-mass
frame,
– frame-2 the Lorentz boosts to bring t and t¯ separately at
rest are defined with respect to the lab frame.
These two frames are designed to be maximally
sensitive to the different polarisation structures of the top-pair
Relative uncertainties
Relative uncertainties
-0.5
0
cos θl+l− in frame-1
0.5
Fig. 14 Same as Fig. 13, but for reference frame-1
in the final state. Furthermore, as already demonstrated at
LO [
37
], considering the spin information in the decay
of the top and anti-top quark is crucial to disentangle the
two different final states, which otherwise look identical,
being characterised by a completely flat distribution in both
cases.
In Fig. 13 we show the behaviour of cos θll in the lab
frame. To highlight shape differences, the background
predictions have been normalised to the inclusive t t¯H cross
section. The upper portions of Figs. 13, 14 and 15 presents the
comparison of the LO and NLO predictions for t t¯H and t t¯γ γ
Cosine θl+l− in the frame-2
]
b
p
[
n
i
b
r
e
p
γ
γ
¯t
t
σ
to tto
t ,
,H γγ
σ¯tt σ¯tt
1.2e-07
]b 1e-07
p
i[n 8e-08
b
r 6e-08
e
p
¯tH 4e-08
t
σ 2e-08
Fig. 15 Same as Fig. 13, but for reference frame-2
separately and shows their respective differential K -factors.
The histograms in the lower portion of the figures
compare results for the signal and background processes, with
their ratio and respective relative uncertainty in the bottom
insets.
This is to be compared with the plots in Figs. 14 and 15,
where the same observable is shown in frame-1 and frame-2.
In the two latter frames a difference in the sign of the slope
emerges, while in the lab frame, despite a clear difference in
the slopes, the curves have an analogous trend. By comparing
the two ratio plots at the bottom of the right plots in Figs. 14
and 15, we conclude that the frame-1 offers the best
signalto-background ratio. It is also worth stressing that, while in
the lab frame the K -factors tend to decrease slightly when
cos θll → 1, in frame-1 and frame-2, the NLO corrections
feature an almost perfectly flat K -factor which agrees with
the inclusive cross section K -factor reported in Table 1. A
comparison of the LO and NLO predictions reveals the
anticipated reduction of the scale uncertainties.
Let us finally remark that the results shown here for the
background, only consider photon radiation from the initial
state and from the top and anti-top quark, but not from their
decay products. This is opposite to the case of the signal,
where photons always originate from the decay of the Higgs
boson. By considering more general cases and using top
tagging techniques without relying on MC truth is expected to
decrease the purity of the signal. A more quantitative analysis
of these effects is beyond the scope of the present work.
4 Conclusions
The event generator MG5_aMC and the one-loop
amplitudes provider GoSam have been interfaced to provide the
user with a framework implenting the most advanced
techniques for the evaluation of cross sections and differential
distributions at next-to-leading order (NLO) accuracy.
In this work, the integration of the two codes has been
applied for the first time to the NLO corrections to the
production of a Higgs Boson in association with a pair of top–
anti-top quarks, as well as to the background process where
two hard photons are produced directly. We compared
several key distributions to disentangle the two processes and
focussed in particular on observables designed to study spin
correlation effects. We found that NLO corrections give a
sizeable contribution, which, however, distorts the shape of
the distributions only very mildly. Moreover, we observed a
clear reduction of the theoretical uncertainties.
The high-level of flexibility and reliability of the joined
technologies of the two codes make of the combination of
MG5_aMC and GoSam an ideal tool for the high-precision
studies and the hunt for deviations from known-physics
signals which characterise the Run II programme at the LHC.
Acknowledgments We thank German Rodrigo and Jan Winter for
valuable discussions. The work of H.v.D., R.F., G.L. and P.M. is
supported by the Alexander von Humboldt Foundation, in the framework of
the Sofja Kovaleskaja Award Projects “Advanced Mathematical
Methods for Particle Physics” (H.v.D., G.L. and P.M.) and “Event Simulation
for the Large Hadron Collider at High Precision” (R.F.), endowed by
the German Federal Ministry of Education and Research. The work of
G.O. is supported in part by the NSF under Grants PHY-1068550 and
PHY-1417354. V.H. is supported by the Swiss National Fund for
Science (SNSF) under grant number P300P2_161050. This research made
use of the CTP computational cluster of the New York City College of
Technology. We thank the CP3 IT team for their constant support and
the availability of the cluster hosted by the Université Catholique de
Louvain.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: The GoSam input card
We report here a copy of the default GoSam input card with a
brief explanation of the different options. For a more detailed
overview we refer to the GoSam papers [
17,20
] and the
GoSam manual which can be found online (http://gosam.
hepforge.org) and is continuously updated. The default
gosam.rc file is the following:
###############################################
# Copy this file to setup.in in order to set
# some common options for all examples.
###################
# physics options #
###################
# Model specs.:
model=smdiag_mad
model.options=GF: 0.0000116639, mZ: 91.188,
mW: 80.419, Nf: 4
# Parameters set to zero algebraically:
zero=me,mmu,mU,mD,mC,mS,wB,wT
# Symmetries:
symmetries=family,generation
# Filter for scale-less loop integrals:
filter.nlo=lambda d: (not (d.isScaleless()))
###################
# program options #
###################
form.bin=tform
form.threads=4
form.tempdir=/tmp
fc.bin=gfortran -O2
###############################################
This input file needs to be modified if the computation
is performed within the 5-flavour scheme. The b-quark mass
can be set algebraically to zero by adding mB to the list zero:
zero=me,mmu,mU,mD,mC,mS,mB,wB,wT.
Furthermore the number of light quarks Nf has to be set equal
to 5.
The tag symmetries specifies some further symmetries
in the calculation of the amplitudes. The information is used
when the list of helicities is generated. Possible options are:
– flavour: does not allow for flavour-changing
interactions. When this option is set, fermion lines are assumed
not to mix.
– family: allows for flavour-changing interactions only
within the same family. When this option is set, fermion
lines 1-6 are assumed to mix only within families. This
means that e.g. a quark line connecting an up with down
quark would be considered, while a up-bottom one would
not.
– lepton: means for leptons what “flavour” means for
quarks.
– generation: means for leptons what “family” means
for quarks.
Furthermore it is possible to fix the helicity of particles. This
can be done using the command %<n>=<h>, where < n >
stands for a PDG number and < h > for an helicity. For
example %23=+- specifies the helicity of all Z -bosons to be
“+” and “−” only (no “0” polarisation).
The filter
filter.nlo=lambda d:
(not (d.isScaleless()))
removes possible scaleless loop diagrams which may be
generated by QGRAF. Several predefined filters to select only
subsets of diagrams exist and can be used in this tag. The
full list and some examples can be found in http://gosam.
hepforge.org.
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